Mechanical Engineering - Theses
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ItemA Framework for Multidimensional Analysis and Development of Numerical Schemes( 2020)Partial differential equations are found throughout engineering and sciences. Under the constraint of complex initial and boundary conditions, most of these complex equations do not have analytical solutions, and therefore require solution by numerical methods. In the context of this thesis, the goal is to examine the governing equations of fluid mechanics (Euler and Navier Stokes) which require both spatial and temporal discretization. Under the effects of numerical differencing, the numerical solution is subjected to both dispersion and dissipation error. These error can be identified and analyzed through spectral analysis method. The analysis of numerical schemes under a coupled spatial temporal framework in one dimensional wavespace is well understood. However, the extension of these methods to multidimensional wavespace and the spectral properties of a hybrid finite difference/Fourier spectral spatial discretization method in multidimensional space is not well understood. Furthermore, the extension of this multidimensional analysis framework to non-linear shock capturing schemes is not done before. This dissertation introduces a generic method for the spectral analysis of linear and non linear finite difference schemes in multidimensional wavenumber space. The aim is to understand the properties of the coupled system for a series of representative spatial and temporal schemes. Theoretical predictions are then compared with numerical solutions based on model equations such as the advection, linearized Euler and linearized Navier Stokes equations. Finally, this framework is used to develop a spectrally optimized hybrid shock capturing scheme which switches between a linear and non linear scheme. Various canonical numerical examples were conducted in order to compare the spectral properties of the new scheme with existing numerical schemes. For the one dimensional linearized Euler equation, it was shown that the dispersion relation belonging to the largest eigenvalue provides the limiting criteria for the stability limit as well as the onset of dispersion error. When the linear spectral analysis method is extended to the two dimensional wavespace, the dispersion and dissipation properties of the coupled schemes become a function of both the reduced wavenumber and the wave propagation angle. When the two dimensional linear spectral analysis method is extended to the two dimensional linearized Compressible Navier Stokes equations (LCNSE), viscous and acoustic effects are taken into account in addition to the convection effects. The addition of the acoustic term to the dispersion relation leads to a coupling of the resolution characteristic such that the group velocities in either spatial direction become a function of the wavenumber in both spatial directions. The two dimensional spectral analysis method was extended to non linear finite difference schemes based on a quasi-linear assumption. In this assumption, the contribution of the harmonic modes (as a result of the non linear differentiation) are neglected during the calculation of the modified wavenumber of the spatial scheme. Using the semi-discretized dispersion relation of the two dimensional advection and linearized Euler equations, the dispersion and dissipation property of a non linear scheme in two dimensional wavespace can be quantified. Using this framework, a non linear scheme, HYB-MDCD-TENO6 was developed based on the principle that the linear part of the scheme can be optimized for minimum dispersion and dissipation error. Furthermore, the non linear part of the scheme is only activated in the vicinity of a sharp gradient. Through a series of numerical experiments, it was found that the hybrid scheme optimized based on the linearized Euler equation tend to give slightly better results than the one optimized based on the advection equation in some of the numerical experiments. In all cases, it was found that the HYB-MDCD-TENO6 scheme provides better resolution than existing baseline TENO and WENO-JS schemes for the same grid size considered.